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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dvdmsscn | Structured version Visualization version GIF version | ||
| Description: 𝑋 is a subset of ℂ. This statement is very often used when computing derivatives. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
| Ref | Expression |
|---|---|
| dvdmsscn.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
| dvdmsscn.x | ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) |
| Ref | Expression |
|---|---|
| dvdmsscn | ⊢ (𝜑 → 𝑋 ⊆ ℂ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | restsspw 17476 | . . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) ⊆ 𝒫 𝑆 | |
| 2 | dvdmsscn.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) | |
| 3 | 1, 2 | sselid 3981 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆) |
| 4 | elpwi 4607 | . . 3 ⊢ (𝑋 ∈ 𝒫 𝑆 → 𝑋 ⊆ 𝑆) | |
| 5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
| 6 | dvdmsscn.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
| 7 | recnprss 25939 | . . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
| 8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
| 9 | 5, 8 | sstrd 3994 | 1 ⊢ (𝜑 → 𝑋 ⊆ ℂ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ⊆ wss 3951 𝒫 cpw 4600 {cpr 4628 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 ↾t crest 17465 TopOpenctopn 17466 ℂfldccnfld 21364 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-rest 17467 |
| This theorem is referenced by: dvxpaek 45955 etransclem17 46266 etransclem18 46267 etransclem20 46269 etransclem21 46270 etransclem22 46271 etransclem29 46278 etransclem31 46280 etransclem34 46283 etransclem43 46292 etransclem46 46295 |
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